Computation of Sample Mean Range of the Generalized Laplace Distribution

A generalization of Laplace distribution with location parameter $\theta$, $\, -\infty 0,$ is defined by introducing a third parameter  $\alpha>0$ as a shape parameter. One tractable class of this generalization arises when $\alpha$ is chosen such that 1/$\alpha$ is a positive integer. In this article, we derive explicit forms for the moments of order statistics, and mean values of the range, quasi--ranges, and spacings of a random sample corresponded to any member of this class. For values of the shape parameter $\alpha$ equal $1/i, i=1,\dotsb,8$, and sample  sizes equal 2(1)15 short tables are computed for the exact mean values of the range, quasi--ranges, and spacings. Means and variances of all order statistics are also tabulated.


Introduction
The probability density function of Laplace distribution with location parameter  ,       , and scale parameter 0   , takes the form This distribution is also known in the literature as the double exponential distribution. Laplace distribution has been extensively studied since its early introduction by Laplace in (1774). For extensive reviews of this distribution and its statistical properties we refer the reader to Johnson et al. (1995) and Kotz et al. (2001).
One possible generalization of Laplace distribution is to introduce a shape parameter ,  0   . In such case, the probability density function (1.1) takes the form ( by varying the value of  , the model can better fit data with sharper peaks and heavier tails. The generalized Laplace distribution can therefore be viewed as a flexible model able to cope with empirical deviations from the Laplace model. One interesting class of (1.2) is when the shape parameter  is assumed to be equal to 1 / , 1,2, ... ii  , i.e., when 1/ is a positive integer value. As it is known that when  is a positive integer, ( , )   can be expanded as a finite summation (Gradshteyn and Ryzhik 2007) be the corresponding order statistics of a random sample of size ( 2)  The sample range, the symmetric quasi-ranges, and the spacings and their linear combinations have many theoretical uses and applications in statistical inference. For excellent surveys of their theoretical uses we refer the reader to (David andNagaraja 2003, Arnold 2008). Applications of order statistics, in general, and the sample range and quasi-ranges, in particular, in estimation and tests of hypotheses are extensively surveyed in (Harter andBalakrishnan 1996, 1998). (For an early survey of such applications, see also (Chu 1957). Use of sample quasi-ranges in estimating population standard deviation is studied by Masuyama (1957), Harter (1959), and Leone et al. (1961). David and Nagaraja (2003) indicated the uses of spacings in building nonparametric and simeparametric confidence intervals for the population parameters, and Ahsanullah et al.
(2013) covered their uses in quantile estimations. The generalized Laplace distribution on the other hand, is employed in Bayesian inference and modeling (Taylor 1992), and its applications in communications, economics, engineering, and finance are extensively surveyed in (Kotz et al. 2001).
Using the expressions of the density and distribution functions in (1.2) and (1.5), respectively, exact explicit forms for the density function of the th r order statistic , r 1,..., r Xn  , and the th s moment of r X about the location parameter ,  are derived in the next section. The means of the range, quasi-ranges, and spacings of samples taken from the distribution under consideration are the subject of section 3. Computational aspects and directions on statistical applications of the derived means are considered in section 4. The computed tables for samples of sizes 2,3,…,15 are given in the appendix.

Moments of order statistics
The following two propositions derive the probability density function of the th r order statistic r X , and the th s moment of r X about the location parameter ,  respectively.
Proof: The probability density function of the th r order statistic is given by Hence, for , x   by substituting from (1.2) and (1.5) in (2.5), and following similar steps as before, we reach the lower right hand side of (2.1).
Deriving the distribution function of r X is a straightforward exercise involving simple variable transformations and some algebraic manipulation. The resulting distribution comprises two functions of finite series of the complementary incomplete gamma function.

Proposition 2.2
Under the same stated assumptions of proposition 2.
Where for the non-negative integer values 0 1 ( and k n r G  is similarly given by (2.7) but with ( ) k n r  in place of ( 1) kr  . Proof: Consider the th s central moment of the th r order statistic, After substitution from (2.1), and making the change of variable where k n r G  is as given by (2.7) but with ( ) k n r  in place of ( 1) kr  . The proof is then concluded by the substitution from (2.9) and (2.10) in (2.8).

Corollary 2.1: On setting
1,   we get the th s moment about the location parameter  of the th r order statistic connected with a sample of size ( 1) n  from the classical twoparameter Laplace distribution (1.1) as This same result appeared in Johnson et al. (1995, p. 168) but with an excessive multiplication by 1 2  .

Means of sample range, quasi-ranges, and spacings
In the early beginning of the last century, Pearson (1902) proved that the expected value of the sample spacings takes the form By summing both sides of (3.1) for all successive spacings from The left hand side of (3.2) represents the mean of the difference between any two order statistics 2 r X and 1 r X . Setting 1 1 r  and 2 , rn  and making use of the binomial theorem, we get the known form of the sample mean range derived by Tippett (1925) Applying the integration by parts reversely on (3.4), we get Another approach to derive (3.5) is to consider the difference between means of the two extreme order statistics 1 and The following theorem states the derivation of explicit forms for the exact mean values of the range, quasi-ranges, and spacings of a random sample taken from the generalized Laplace distribution.
where for the non-negative integer values 0 1 and k n r M  is similarly given by (3.8) but with ( With the change of variable / ( ) z y k r  , the integral in (3.11) is clearly a gamma function, and so   where kr M  is as given by (3.8).
Similarly, the second integral in (3.9), after substituting for ) x ( and then substitute in (3.7) to get both sides of (3.14), the equality of the two sides becomes evidently true.

Computation algorithm and tabulation
Apart from the precision matters of computing large factorials and gamma functions, the main difficulty in programming the derived forms (2.6) and (3.7) is the generation of all possible terms of the involved multinomial expansions. The following algorithm provides a straightforward approach to overcome this difficulty. The following algorithm; in Pascal-like notation, is adopted to generate all of the compositions needed for computing summation like (3.8

Algorithm composition (k + r);
A Fortran 95 code, running on a Windows-based Intel® 3.00GHz CPU is written to conduct all computations. The Fortran language provides better control over the precision and accuracy of the computations. Interfacing this Fortran code with R is now under consideration by the author besides other computational routines attributed to the Generalized Normal and the multinomial distributions.

Tabulation
Given that the sampling distribution is the generalized Laplace (1.2) with parameters 1   and  equals 1, 1/ 2 ,..., 1/ 8 respectively, exact mean values of the range, quasiranges, and spacings, as well as means and variances of all order statistics are computed for samples of sizes equal to 2,3,…,15. Table 1 in the appendix gives the means of ranges and quasi-ranges, and table 2 gives the means of the corresponding lower order spacings. It should be noted that, for any sample size 3 n  , the identity relation (3.14) eliminates the need to compute the means of the ( 1) Govindarajulu (1966), has derived closed-form expressions for the first two moments and the product (linear) moment of order statistics, in terms of the moments of order statistics of a random sample drawn from the negative exponential, as the folded distribution of the double exponential. For samples of size 2,3,...,20, n  Govindarajulu evaluated and tabulated the means of order statistics, accurate to six decimal places, and the covariances, accurate to five decimal places. All his computed means and variances for 2,...,15, n  match exactly the corresponding computed values for this case in tables 3 and 4.

Conclusion
In this article, a generalization of the Laplace distribution is considered. The model enhances the classical Laplace distribution by introducing a third parameter 0,   to further control the shape of the distribution. By varying the value of  , the model can better fit data with sharper peaks and heavier tails. The generalized Laplace distribution can therefore be viewed as a flexible model able to cope with empirical deviations from the Laplace model with two parameters only.
Under the assumption that the reciprocal of the shape parameter is a positive integer, we derive explicit forms for density functions and the moments of order statistics, and the mean values of the range, quasi-ranges, and spacings of a random sample corresponded